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# Rectifiability of Optimal Transportation Plans

Published:2011-11-03
Printed: Aug 2012
• Robert J. McCann,
Department of Mathematics, University of Toronto, Toronto, ON M5S 2E4
• Brendan Pass,
Department of Mathematics, University of Toronto, Toronto, ON M5S 2E4
• Micah Warren,
Department of Mathematics, Princeton University, Princeton, NJ, USA 08544
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## Abstract

The regularity of solutions to optimal transportation problems has become a hot topic in current research. It is well known by now that the optimal measure may not be concentrated on the graph of a continuous mapping unless both the transportation cost and the masses transported satisfy very restrictive hypotheses (including sign conditions on the mixed fourth-order derivatives of the cost function). The purpose of this note is to show that in spite of this, the optimal measure is supported on a Lipschitz manifold, provided only that the cost is $C^{2}$ with non-singular mixed second derivative. We use this result to provide a simple proof that solutions to Monge's optimal transportation problem satisfy a change of variables equation almost everywhere.
 MSC Classifications: 49K20 - Problems involving partial differential equations 49K60 - unknown classification 49K6035J96 - Elliptic Monge-Ampere equations 58C07 - Continuity properties of mappings